q-DEFORMED 𝒲-ALGEBRAS AND ELLIPTIC ALGEBRAS*)

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Abstract

The elliptic algebra 𝒜q,p(Symbol 1(N)c) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the 𝒲N algebra, are constructed. The operators t(z) also close an exchange algebra when (−p1/2)NM = q−c−N for M ∈ Symbol 2. It becomes Abelian when in addition p = qNh where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed 𝒲N algebras depending on the parity of h, characterizing the exchange structures at p ≠ qNh as new 𝒲q,p(sl(N)) algebras.

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